## 1.3.4 Definition of Multi-Step Methods

Measurable Outcome 1.5, Measurable Outcome 1.8, Measurable Outcome 1.13, Measurable Outcome 1.15

The class of finite difference methods known as multi-step methods is one of the most widely-used approaches for solving ordinary differential equations, and forms the basis for solving partial differential equations as well.

## Definition of Multi-Step Methods

The generic form of an \(s\)-step multi-step method is,

A multi-step method with \(\beta _0 = 0\) is known as an **explicit** method since in this case the new value \(v^{n+1}\) can be determined as an explicit function of known values (i.e. from \(v^ i\) and \(f_ i\) with \(i \leq n\)). A multi-step method with \(\beta _0 \neq 0\) is known as an **implicit** method since in this case the new value \(v^{n+1}\) is an implicit function of itself through the forcing function, \(f^{n+1} = f(v^{n+1},t^{n+1})\). We defer implicit methods to Section 1.7.1 and focus on explicit methods here.

**Exercise 1** What are the non-zero \(\alpha _ i\) and \(\beta _ i\) for the forward Euler method?

Using the notation given above, the forward Euler method is:

**Exercise 2** What are the non-zero \(\alpha _ i\) and \(\beta _ i\) for the Midpoint method?

Using the notation given above, the midpoint method is: